Which of these two lenses is better for the Nikon D3100?

I just recently got the Nikon D3100 and have the 18-55mm lens. The camera is great, i just am getting a lens for different events. I need to be able to take pictures from long distances, and sports. The 55-200 lens is more expensive, but has auto focus. The 70-300 is cheaper and more distant, but does not have an auto focus. This is my first DSLR camera but I'm a fast learner. I just need advice. Which lens would be the best for taking sports and other distant pictures? I don't really know a lot about manual focusing, but I can learn.
The link for the 55-200 is: http://www.amazon.com/...000O161X0/
The link for the 70-300 is: http://www.amazon.com/...00005Y3OM/

Added (1). These lenses fit my budget. I can't get the better versions of these. That being said, I guess the question is if I used the 70-300mm lens with no VR and manual focus, will I be able to take photos of sports and other events just as good as the 55-200mm lens with VR and AF?

You need to find the test results for the modulation transfer function for these two lenses to see which produces sharper results. For sports, you probably don't need to worry too much about auto focus since usually you will be setting the focus, and closing down the aperture as much as possible to extend the depth of focus.
One thing you should find for the auto-focus lens is how fast it will focus IF you are going to be shooting at different distances in rapid succession. I got to the method above because of the frustration of slow auto focus.

The other Nikkor AF-S 70-300 mm VR is a much better lens, but you can probably tell that in the differences in price.

If you do not have the budget for the 70-300 VR, get the 55-200 mm instead.

You should really be comparing the AF-S 70-300mm and the AF-S 50-200mm lenses.

Look at this webpage to see the difference between 200 and 300mm, and then decide if the 300mm is worth $200 more.

http://www.althephoto.com/concepts/lenses.php